SUMMARY
The discussion centers on the claim that no natural number \( n \) can satisfy the condition \( n \equiv 1 \mod p_i \) for all primes in a sufficiently large set \( \{ p_{1}, p_{2}, \dots, p_{h} \} \). The counterexample provided demonstrates that choosing \( n = p_1 p_2 \cdots p_h + 1 \) indeed satisfies the condition for any set of primes, not necessarily consecutive. This proves that the initial assertion is incorrect, as the existence of such an \( n \) is guaranteed by the properties of modular arithmetic.
PREREQUISITES
- Understanding of modular arithmetic and congruences
- Familiarity with prime numbers and their properties
- Basic knowledge of number theory concepts
- Ability to manipulate algebraic expressions involving primes
NEXT STEPS
- Study the properties of modular arithmetic in depth
- Explore the concept of prime factorization and its implications
- Learn about the Chinese Remainder Theorem and its applications
- Investigate advanced number theory topics, such as Dirichlet's theorem on arithmetic progressions
USEFUL FOR
Mathematicians, students of number theory, and anyone interested in the properties of prime numbers and modular arithmetic.